Stochastic Integrals and Their Expectations

# Introduction

This article describes the *Mathematica* package *ItoIntegralRules *that provides facilities to simplify and compute expectations of stochastic integrals. *ItoIntegralRules* extends the previous package *Itovsn3* [1, 2] that implements stochastic calculus within *Mathematica*.

Stochastic calculus is famous for providing the foundations for modern mathematical finance and is also used extensively in a large number of other areas of applied probability. The introductory text by Øksendal [3] strikes an excellent balance between theory and accessibility. Here we give a very brief review of the underlying concepts. A central notion for stochastic calculus is that of a (continuous) *semimartingale*: a random process *X* that can be written as the sum of a *local martingale* *M* (for example, Brownian motion) and a *drift process* *V* (a continuous process of locally bounded variation, typically the solution of some conventional differential equation). The decomposition is unique and can be thought of as a decomposition of *X* into signal *V* plus noise *M*. Fundamental to the theory of stochastic calculus is *Itô's lemma*: if is a smooth function of the semimartingale *X*, then

where is the quadratic variation, the unique nondecreasing process such that is a local martingale begun at 0. In the case when *M* is Brownian motion, we find . Care has to be taken when interpreting the integral : nontrivial continuous local martingales *M* do not possess bounded variation, so the component of must be interpreted as a *stochastic* or *Itô integral*. (Since *V* is of locally bounded variation, the interpretation of is strictly classical.)

Itô's lemma, in conjunction with martingale theory, permits us to calculate effectively with semimartingales in *Mathematica*. In *Itovsn3* [1, 2] the underlying algebra of stochastic calculus is implemented as an algebra of *stochastic differentials* *dX*, *dM*, and *dV*. This has facilitated several investigations into applied probability problems: examples given in [2] include explorations of the statistical theory of shape, coupling of diffusions, and computation of distributions of special random processes. The underlying principle of *Itovsn3* is to recognize a second-order algebraic structure of differentials corresponding to the formula for Itô's lemma. Thus semimartingales *X* have stochastic differentials *dX* that can be multiplied together to obtain a differential measure of volatility of the semimartingale (for example, ) and that possess drift parts that capture the underlying trend ( if ).

The need to perform some calculations related to an image analysis problem [4, 5] supplied the initial motivation to extend *Itovsn3* by adding the package *ItoIntegralRules* to more fully implement a notion of *Itô integral,* such as or . Itô integrals are represented in *Itovsn3* using placeholders ItoIntegral[g dM] that possess the bare minimum of properties (loosely speaking, ItoIntegral[dM] -> M). The new package *ItoIntegralRules* adds facilities to simplify expressions involving ItoIntegral in various ways and also adds an expectation operator . In particular, this allows us to address the calculations arising from the image analysis problem, which requires the derivation and further manipulation of formulas for means and variances for integrated Ornstein-Uhlenbeck processes. These specific calculations can of course be performed directly by hand; however, the computational framework provided by *ItoIntegralRules* covers a much wider range of possible calculations, so it should be of use elsewhere.

This article is divided into three sections: the first summarizes the issues of simplification of expressions involving ItoIntegral, the second introduces a notion of expectation and its interaction with ItoIntegral, and the conclusion discusses possibilities for further work.

## Related Work

There are other implementations of stochastic calculus within a computer algebra package. Steele and Stine [6] adopt a diffusion-based approach, which has been developed further by Mark Fisher in the ItosLemma.m package [7]. Cyganowski [8] describes an approach using Maple, including a solver for stochastic differential equations.

## Installation

The installation of *Itovsn3* and *ItoIntegralRules* follows the usual procedure for *Mathematica* packages. Unpack the zip archive file Itovsn3.zip (see Additional Material) either in the current working directory or in the Applications subdirectory of *Mathematica*'s AddOns directory (in the second case the packages will load no matter what is the current working directory). This will place the files init.m (which contains the package *Itovsn3*), ItoIntegralRules.m, and ItoIntegralTests.nb in the Itovsn3 subdirectory. The accompanying notebook, ItoIntegralTests.nb, contains detailed examples and unit tests for *ItoIntegralRules*.

After installation, *Itovsn3* can be loaded and initialized and a single Brownian motion *B* can be introduced by